Improved Backpropagation Algorithms by Exploiting Data ...jpier.org/PIERB/pierb66/01.15120204.pdf · Improved Backpropagation Algorithms by Exploiting Data Redundancy in Limited-Angle

Progress In Electromagnetics Research B, Vol. 66, 1–13, 2016

Improved Backpropagation Algorithms by Exploiting DataRedundancy in Limited-Angle Diffraction Tomography

Pavel Roy Paladhi1, *, Ashoke Sinha2, Amin Tayebi1, Lalita Udpa1, and Satish Udpa1

Abstract—Filtered backpropagation (FBPP) is a well-known technique used in DiffractionTomography (DT). For accurate reconstruction of a complex-valued image using FBPP, full 360◦ angularcoverage is necessary. However, it has been shown that by exploiting inherent redundancies in theprojection data, accurate reconstruction is possible with 270◦ coverage. This is called the minimal-scanangle range. This is done by applying weighting functions (or filters) on projection data of the object toeliminate the redundancies. There could be many general weight functions. These are all equivalent at270◦ coverage but would perform differently at lower angular coverages and in presence of noise. Thispaper presents a generalized mathematical framework to generate weight functions for exploiting dataredundancy. Further, a comparative analysis of different filters when angular coverage is lower thanminimal-scan angle of 270◦ is presented. Simulation studies have been done to find optimum weightfilters for sub-minimal angular coverage. The optimum weights generate images comparable to a full360◦ coverage FBPP reconstruction. Performance of the filters in the presence of noise is also analyzed.These fast and deterministic algorithms are capable of correctly reconstructing complex valued imageseven at angular coverage of 200◦ while still under the FBPP regime.

1. INTRODUCTION

Diffraction tomography (DT) is a popular imaging modality [1, 2] used in a variety of applications suchas medical imaging, non-destructive evaluation of materials, structural health monitoring, geophysicsetc. [3, 4]. In the domain of optical imaging, this method has been explored in depth viz. opticaldiffraction tomography (ODT) with multi-disciplinary applications [5–7]. DT is a broad imagingtechnique of which ultrasound and microwave tomographic imaging are sub-classes. It is a comprehensiveway of characterizing the complex valued object-function of the test object. The scheme is applicableto microwave tomography of tissue samples. It has been of great interest in tomography of humanbreast to identify malignant tumours [8–14]. The high contrast between healthy and cancerous breasttissue provides opportunities for clearly identifying malignancies within breast tissue. The contrast ismuch higher than in case of X-rays and hence microwave tomography is advantageous both in terms oftrue detection and radiation damage from X-rays. Further applications are in fields of SAR imaging.Various backprojection techniques have been explored for use in radar imaging and modified for betterreconstructions [15–17]. Faster implementations for backprojections algorithms are also being intenselyexplored, e.g., [17]. Again, radar based methods have been combined with microwave tomographyto generate higher resolutions for medical imaging [10]. Thus, any improvement in the traditionalbackpropagation techniques is of potential interest across multiple disciplines. This paper shows anapproach to improve direct FBPP reconstruction of complex-valued objects at lower angular coveragesthan traditional requirements.

Received 2 December 2015, Accepted 9 February 2016, Scheduled 9 February 2016* Corresponding author: Pavel Roy Paladhi ([email protected]).1 Department of Electrical and Computer Engineering, Michigan State University, 428 S Shaw Lane, Rm-2120, East Lansing, Michigan48824, USA. 2 Department of Statistics and Probability, Michigan State University, 619 Red Cedar Road, C413 Wells Hall, EastLansing, Michigan 48824, USA.

2 Roy Paladhi et al.

In the presence of weak scatterers, assuming the Born or Rytov approximations [2], the FourierDiffraction Projection theorem (FDP) is applied. The FDP relates the scattered field data fromthe Region of Interest (ROI) to the 2D-Fourier Space of the ROI. Devaney developed the filteredbackpropagation (FBPP) algorithm for this configuration to reconstruct a low-pass filtered image ofthe object function [1]. The traditional backpropagation technique requires projection data from [0, 2π]angular coverage for accurate image reconstruction of a complex image. Even though the formulationapplies to weak scatterers, it has found a number of applications and since its introduction, a steadyresearch focus has been maintained to increase the extent of usage for backpropagation-like algorithmsfor DT and making them more general in applicability, e.g., [18, 19]. A major challenge in manyreal world situations is that projection measurements cannot be gathered over full 360◦ view aroundthe test object. With limited access to the object and decrease in angular coverage, the availabledata in the Fourier space decreases. Reconstruction from this partial Fourier space data leads tomany artifacts and loss of important image features. Hence alternate schemes are needed for imagereconstruction from limited angular coverage projection data. For highly sparse data or limited angularaccess various minimization, regularization, estimation techniques and statistical approaches are beingsuccessfully explored to make the reconstruction algorithms more robust to sparse and noisy data,e.g., [20, 21]. However, generally these are iterative methods and the total computation time dependson the convergence rate of the algorithm. A single step method would be always quicker provided it canhandle the limited availability of data. In [23–26], a novel alternative approach for moderately limitedangular access was introduced. It was shown that using inherent redundancies in the projection datafrom the conventional setup, exact reconstruction is possible with data from [0, 3π/2] coverage. For lowerangular coverages, the algorithm results in significant distortions. In this paper we explore a techniquethat efficiently uses the redundancies in the Fourier space data from the conventional setup [27]. Thistechnique can reconstruct better images (than regular FBPP) effectively over any range between πto 3π/2. This is a crucial development, as the demands on angular access for complex valued objectreconstruction is lowered considerably. This paper proposes to use projection datasets optimally to getenhanced reconstructions from lower angular coverages. Also, this is a direct reconstruction methodwhich does not employ any error minimization algorithms and hence is faster and accurate over angularcoverages between 180◦–270◦.

The paper is organized as follows: Section 2 gives a short background of the FDP and FBPPalgorithms and explains the minimal-scan complete dataset proposed in [23] and how equivalent systemsof backpropagation algorithms can be generated. Section 3 introduces methods to generate weightfunction classes of backpropagation algorithms which can give better reconstruction than regular FBPPin the angular coverage range where the redundancy can still be exploited, i.e., between 180◦–270◦.Section 4 presents results from different backpropagation classes and analyses the relative performancesat angular coverages below 270◦.

2. FBPP & THE MINIMAL SCAN REQUIREMENT IN DT

The fundamental theory underlying 2D-DT is the FDP. This relates the scattered field data fromthe Region of Interest (ROI) due to incident plane waves to the 2D-Fourier space of the ROI [2].The traditional 2D-configuration is shown in Fig. 1. Let the object o(x, y) be illuminated with amonochromatic plane wave of frequency ν0. Let the wave be incident at an angle φ to the horizontalaxis. Then, the 1D Fourier Transform (FT) of the scattered field measured along the straight line η = lin the co-ordinate system (ξ, η) gives the values of the 2D transform of the object O(νx, νy) along asemi-circular arc in the frequency domain. As shown in the right half of Fig. 1, this arc will be tiltedat an angle φ. The scattered field data and the object function are related by the following equation:

U(ν, l) =j

2√

ν20 − ν2

ej√

ν20−ν2lO

(ν,

√ν20 − ν2 − ν0

), (1)

where U(ν, l) represents 1D FT of the scattered field, u(ξ, η) under Born approximation (measured atline η = l), and ν lies in the range [−ν0, ν0].

Devaney developed the well-known filtered backpropagation method in [1]. In polar coordinates,

Progress In Electromagnetics Research B, Vol. 66, 2016 3

(a) (b)

Figure 1. (a) Classical scan configuration of 2D DT and (b) relation of scattered field data with the2D Fourier space of the objective function.

as presented in [23], the backpropagation integral takes the form:

a(r, θ) =∫ 2π

φ=0

∫ ν0

νm=−ν0

ν0

ν ′ |νm|M(νm, φ) · e[j2πνm cos(φ−α−θ)]dνmdφ (2)

In Eq. (2), a(r, θ) is the objective function being reconstructed in polar spatial coordinates (r, θ),ν0 the frequency of the incident monochromatic plane wave and φ being the incidence angle, M(νm, φ)a modified 1D FT of the scattered data, defined as

M(νm, φ) = U(νm, φ)jν ′

2π2ν20

e−j2πν′l, (3a)

where ν ′ =√

ν20 − ν2

m, νa = sgn(νm)√

ν2m − ν2

μ, νμ = j(ν ′ − ν0) and

α = sgn(νm) arcsin(

12ν0

√ν2

m − ν2μ

), (3b)

Without loss of generality, α could be further simplified to

α =12

arcsin(

νm

ν0

). (3c)

In order to reconstruct a complex object accurately, a full knowledge of M(νm, φ) in the range [0, 2π]is necessary to perform the integration in Eq. (2), [22].

2.1. Minimal Scan FBPP

The Fourier space redundancies from standard 2D-DT projection data and reconstruction method fromlower coverage utilizing the redundancy are now explained. Consider the two arcs OA and OB in Fig. 1.The two arcs individually traverse the transform space (i.e., the Fourier space) as the interrogatingwave angle changes between [0, 2π]. Now consider their traversals in the Fourier space individuallyfor angular coverage of [0, 3π/2]. This is illustrated in Fig. 2 below. Individually each half does anincomplete traversal of the Fourier space as shown in Figs. 2(a) and 2(b). However, if superimposed, asseen in Fig. 2(c), the entire Fourier space is covered with some areas of overlap. The key point is that at270◦ there is in effect, a complete coverage of the Fourier space. This scan range of [0, 270◦] is referredto as the minimal scan angle and is the minimal angular coverage required for exact reconstruction [23].

To better appreciate the principle, the Fourier space domain is re-plotted in a modified coordinatesystem, where the spatial frequency is plotted along x-axis and the angular coverage along y-axis.Physically this new coordinate system can be viewed as ‘straightening’ the arcs AOB from each


(a) (b)

(d)(c)

Figure 2. (a) Fourier space coverage for 270◦ angular access by segment OB in Fig. 1(b) same forsegment OA of Fig. 1. (c) Superposition of the two coverages from (a) and (b). (d) The Fourier data-space in alternate co-ordinate system with angular coverage along y-axis and the wave number alongx-axis.

projection and stacking them up on top of each other sequentially. In this layout, the entire Fourierdataspace can be divided into four sub-regions A, B, C and D as shown in Fig. 2(d). The boundariesfor the four regions can be expressed as A = [|νm| ≤ ν0, 0 ≤ φ < 2α + π/2], B = [|νm| ≤ ν0, π/2 + 2α ≤φ < 2α + π], C = [|νm| ≤ ν0, π + 2α ≤ φ < 3π/2] and D = [|νm| ≤ ν0, 3π/2 ≤ φ < 2π]. As seen in (2c),α is a function of νm and so the regions have nonlinear boundaries.

From FDP theorem the following periodicity holds [24]:

M(νm, φ) = M(−νm, φ + π − 2α) (4)

This also implies that for every point in region A (or B), there is a point of identical value in region C(or D). Thus, the knowledge of M(νm, φ) in regions A and B, makes information in C and D redundant.This redundancy can be handled by normalizing the dataspace using appropriate weight filters. Theweighted dataset M ′(νm, φ) can be defined as M ′(νm, φ) = w(νm, φ)M(νm, φ), where w(νm, φ) satisfiesthe condition

w(νm, φ) + w(−νm, φ + π − 2α) = 1. (5)

It should be noted that since region D is unavailable in a 270◦ coverage we set w(νm, φ) = 0 in regionD and correspondingly w(νm, φ) = 1 in region B. For regions A and C, any weight functions arevalid as long as they satisfy Eq. (5). Using the weighted dataset M ′(νm, φ) we can apply the regularbackpropagation algorithm for reconstruction. This is called Minimal-Scan Filtered Backpropagation(MS-FBPP) which involves evaluating the following integral

aW (r, θ) =12

∫ 3π/2

φ=0

∫ ν0

νm=−ν0

ν0

ν ′ |νm|M ′(νm, φ) · e[j2πνm cos(φ−α−θ)]dνmdφ. (6)


This integral can, in theory, exactly reconstruct an image from a 270◦ angular coverage by utilizingdata redundancy in projection data. The weights w(νm, φ) can then be used to define classes ofbackpropagation algorithms for image reconstruction. An example of weight functions introduced in [23]is given in equation below:

w(νm, φ) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

sin2

[π

4· φ

π/4 + α

], in A,

1, in B,

sin2

[π

4· 3π/2 − φ

π/4 − α

], in C,

0, in D.

(7)

This will be used as a reference later on in Section 4 for comparison with the weight functions that areproposed in this paper. The filter of Eq. (7) will be referred to as sine-squared filter in this paper. This isa very elegant and simple filter designed to have continuity across the boundaries between regions A, B,C and D. To demonstrate the efficacy of MS-FBPP, a sample reconstruction is performed on a Shepp-Logan type phantom with complex parameter distribution. The real part of the phantom is shown inFig. 3(a) and the imaginary part in Fig. 4(a). Reconstruction from regular FBPP and MS-FBPP usingweights of (7) are given in Fig. 3 and Fig. 4. The images show the MS-FBPP algorithms capable ofgenerating accurate reconstructions from 270◦ (equivalent to full coverage), whereas the regular FBPPimage shows considerable distortion at 270◦.

Weight filters which are discontinuous across boundaries of the four regions in Fig. 2(d) can giverise to artifacts, especially in the case of discrete data. Further, for each class of weights, its distributionin frequency space of Fig. 2(d) also determines the performance when the available coverage is below270◦. This is because, C and D are complementary to regions A and B respectively. An efficient weightfunction set would be that which spans most of the regions A and B, thus limiting the requirement toaccess regions C and D. In effect such weights can generate good reconstructions even from angularcoverages below 270◦. The next section explains a systematic approach to generate general classes ofweight functions.

(a) (b)

(d)(c)

Figure 3. Demonstration of the MS-FBPPconcept, (a) real part of original image, (b) FBPPreconstruction from 360◦ coverage, (c) FBPPreconstruction from 270◦ coverage and (d) MS-FBPP reconstruction from 270◦ coverage.

(a) (b)

(c) (d)

Figure 4. Demonstration of the MS-FBPPconcept, (a) imaginary part of original image,(b) FBPP reconstruction from 360◦ coverage, (c)FBPP reconstruction from 270◦ coverage and (d)MS-FBPP reconstruction from 270◦ coverage.


3. GENERATION OF EFFICIENT WEIGHT FUNCTIONS

From previous section, we see that the weights in regions B and D are 1 and 0 respectively. Notice thatweights in region C can be generated from weights of region A, because for any point (νm, φ) in region C,the point (−νm, φ+π−2α) in A is equivalent due to (4), and we get w(νm, φ) = 1−w(−νm, φ+π−2α),using Eq. (5). Thus it is sufficient to generate weights for the region A only.

Furthermore from Eq. (5), the non-negative weights are bounded above by 1. So in region A, forany fixed νm, the function w(νm, ·) =: F (·) is defined on [0, 2α + π/2], and takes values between 0 and1. However it is desirable to have weights which are continuous at the boundaries between two regions.Weights which are discontinuous at the boundaries will generate artifacts in case of discrete datasets asnoticed in [23].

Here we propose an approach to generate the weights, by using cumulative distribution functions(cdf) to model F (·), which are guaranteed to be bounded within 0 and 1. To obtain continuous weights,we use continuous F with

F (0) = 0, and F (2α + π/2) = 1. (8)This will ensure that weights are continuous at the boundary between regions A and B, and consequentlyat the boundaries between regions B and C, and C and D. This approach would allow us to chooseweights from a no. of cdfs. In this article, we primarily use beta-cdf to generate weights. The standardbeta-cdf is defined as:

F (x|a, b) =∫ x

−∞f(t|a, b)dt. (9)

where a > 0, b > 0, and f is the standard beta probability density function:

f(t|a, b) =

⎧⎨⎩

1B(a, b)

ta−1(1 − t)b−1, 0 ≤ t ≤ 1,

0, otherwise,(10)

and B(a, b) =∫ 10 ta−1(1 − t)b−1dt. We obtain a family of beta-cdf’s by changing values of a and b

in Eq. (10), as shown in Fig. 5. Notice that the standard beta-cdf has support [0, 1], while in region A

Figure 5. Beta-cdf plots for different values ofthe parameters a and b.

Figure 6. Images showing weight distributions inFourier domain generated by parametric variationof the beta-cdf (using the parameters shown inFig. 5). Frequency is plotted along abscissa andangular coverage along ordinate.


for a fixed νm, we need to define weights for φ ∈ [0, 2α + π/2]. This can be achieved by substitutingx = φ

2α+π/2 in Eq. (9). The corresponding weight profiles in the frequency domain are shown in Fig. 6.The plots in Fig. 5 can be used to understand how the weights will be distributed in Regions A and

C. Fig. 6 gives useful insight on choosing optimal parameter values. For example, with combinationsa = 2, b = 5 or a = 6, b = 1, region A is not well covered, whereas for a = 0.4, b = 6, region A has beenalmost fully covered with near unity weights leaving Region C with mostly near-zero weights. Thiscombination is expected to better utilize data redundancy than the other combinations shown in Fig. 5.This can be further illustrated through Fig. 7 and Fig. 8. Fig. 7 shows some weight distributions inthe Fourier space (in the alternative co-ordinate system). The beta-cdf parameters used to generatethese weights are shown in inset white text. Fig. 8 shows the reconstructed real parts of the images byusing these weights. Reconstruction was performed from sub-minimal angular coverage of 200◦. Theweights which cover region A more completely also give better reconstructions. It should be noted thatespecially for coverages lower than 270◦, we receive less information from C, hence an optimum weightfunction should span most of region A with near unit weightage. Since the weights are continuous acrossthe boundaries of the regions, the transition from zero to unity should be adequate to retain as muchinformation as possible within region A but also not too abrupt to minimize generation of artifacts incase of discrete datasets. In this paper the authors demonstrate the efficacy of using cdf based weightsin image reconstruction at sub-minimal angular coverage. To do this, good parameter values wereheuristically determined in the following manner: for the beta-cdf, a parametric sweep over the rangeof a ∈ [0.2, 3], b ∈ [1, 6] was performed and the corresponding weight distributions were generated.Reconstructions were performed with these weights for a sub-minimal coverage of 200◦. The weightcombination which gave the least distortion in the reconstruction and maintained highest correlationwith the original image was chosen as the optimum set. Within this range, the parameter combinationa = 0.4, b = 6 gave best results and is used for image reconstruction in the next section.

Similar approach can be used to generate weight functions from other cumulative distributionfunctions as well, for instance with the gamma-cdf. However, since the gamma distribution has

(a) (b)

(c) (d)

Figure 7. Images showing weight distributions inFourier domain generated by parametric variationof the beta-cdf. The parameters a and b usedfor each weight distribution are inset in each plot.Frequency is plotted along abscissa and angularcoverage along ordinate.

(a) (b)

(c) (d)

Figure 8. Corresponding reconstructions fromusing the weight distributions in Fig. 7.


unbounded support, a suitable transformation of the argument is required to ensure that the weights areconstructed with continuity properties at the boundaries as discussed above. Details about constructingthe weights with gamma-cdf are described in the Appendix. For this case, the parameters were variedin the range a ∈ [0.1, 3.1], b ∈ [0.1, 3.1]. The best parameter set found from this range was a = 2.1,b = 0.1. In this paper, results from beta and gamma distributions using these optimum parameters foreach distribution will be presented and compared with the results from weights given in Eq. (7).

4. RESULTS

This paper presents results from simulated projection data for DT valid under the Born approximation.The test image is complex. Both the real and imaginary parts of the image are, in effect, modifiedversions of the Shepp-Logan phantom, a standard model used to validate computed tomographyalgorithms. The real and imaginary parts of the test image are shown in Figs. 3(a) and 4(a). Theprojection data from the phantom was computed following [23, 24]. The image matrix is 128 × 128pixels with pixel size of λ/8. The image has an area of 16λ × 16λ. The objective of this paper was todefine a procedure to generate optimum weights which can exploit the redundancy for angular coveragesbelow 270◦ and up to 180◦, where redundancy is still present in the projection data.

4.1. Noiseless Reconstruction

Proceeding in the manner described in the previous section to generate weights, the following optimumparameters for different cdfs were used: for beta-cdf, a = 0.4, b = 6; for gamma-cdf, a = 2.1, b = 0.1.Reconstructed images have been plotted from regular FBPP and MS-FBPP using different weightfunctions for coverages 270◦ and 200◦ (an example of sub-minimal angular coverage). Both real andimaginary parts are shown in Figs. 9 and 10 respectively. The reconstructions show the beta-cdf andgamma-cdf based weights generate a very stable reconstruction even in lower angular coverage of 200◦,with beta-cdf performing slightly better overall.

To compare the performance of different MS-FBPP algorithms quantitatively, we use their Mean-Absolute-Error (MAE) with respect to the original image. The MAE is calculated as the absolutemean pixel-by-pixel difference between the original and reconstructed image: MAE = 1

n

∑ |imgorig(i)−imgrecon(i)|, where imgorig(i) is the ith pixel in the original image and imgrecon(i), the ith pixel in thereconstructed image and n is the total number of pixels in the image. The errors in real and imaginaryparts of image were calculated separately. At 200◦ coverage, for the real part of the image, the beta-cdf based reconstruction had a 30.77% lower error than regular FBPP, while the gamma-cdf basedreconstruction yielded a 29.11% lower error. A similar trend is seen for the imaginary part of the image.The sine-squared based weights give accurate reconstructions at 270◦, but below 270◦ they progressivelydeteriorate. These are clearly not optimum choices for coverage < 270◦. For example, at 200◦ coverage,the quality of reconstruction degrades for both the regular FBPP and sine-squared weighted FBPP.However, the latter has a higher error than the regular FBPP reconstruction (2.46% higher in real partof image). A more detailed error-analysis has been done with noisy data in the next subsection. Below180◦ coverage, the redundancy disappears and using weights alone, in principle cannot produce a betterreconstruction than regular FBPP. So, reconstructed images are not shown for further lower coverage.However, it should be noted that for these lower coverages, reconstruction from un-optimized weightswill generate higher errors than regular FBPP. This is also evident from the sine-squared weights basedreconstructions for sub-minimal coverages. The optimized cdf based weights perform much better thanboth the sine-squared weights and regular FBPP.

4.2. Reconstruction with Noisy Data

Noise is an integral part of any measurement system. In a DT setup, noise may arise from randominhomogeneities in medium or may be introduced by the experimental procedure. To account forthese, noisy reconstruction has been modeled as a stochastic process in literature before [28, 29].Reconstruction from noisy data was necessary to examine the reliability of these algorithms whenapplied to practical systems. The MS-FBPP algorithms are expected to respond to noisy data models


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 9. Reconstructed real part from complextest image under noiseless conditions. The leftcolumn shows reconstruction from 270◦ coverageand right column shows reconstruction from 200◦coverage. (a), (b) using regular FBPP, (c), (d)using sine-sq weights, (e), (f) using gamma-cdfweights, (g), (h) using beta-cdf weights.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 10. Reconstructed imaginary part of thetest image under noiseless conditions. The leftcolumn shows reconstruction from 270◦ coverageand right column shows reconstruction from 200◦coverage. (a), (b) using regular FBPP, (c), (d)using sine-sq weights, (e), (f) using gamma-cdfweights, (g), (h) using beta-cdf weights.

differently. To consider the effect of all noise sources, it was assumed as sufficient to consider a whiteGaussian noise distribution in the scattered field data [24]. An additive white Gaussian noise (AWGN)with different variances has been injected to the analytically computed projection data to give differentnoise levels. To observe the non-uniform propagation of errors under noisy data [24, 29], we comparedthe reconstruction from the weighted MS-FBPP algorithms with regular FBPP using the projectiondata injected with 3-dB AWGN. The reconstructed images from noisy data using beta-cdf weights fordifferent angular coverages in the range of [200◦, 270◦] are given in Fig. 11 and Fig. 12. We use here


(a) (b)

(c) (d)

Figure 11. Reconstruction of real part of imagefrom 3dB awgn projection data using beta-cdfweights for angular coverages, (a) 270◦, (b) 250◦,(c) 220◦, (d) 200◦.

(a) (b)

(c) (d)

Figure 12. Reconstruction of imaginary part ofimage from 3dB awgn projection data using beta-cdf weights for angular coverages, (a) 270◦, (b)250◦, (c) 220◦, (d) 200◦.

Figure 13. MAE calculated at differentcoverages for real part of reconstructed imagewith regular FBPP and MS-FBPP using differentweights.

Figure 14. MAE calculated likewise Fig. 13 forimaginary part of reconstructed image.

weights based on beta-cdf with a = 0.4, b = 6, which earlier gave best results with noiseless data. Thereconstructions show robustness of the algorithm to noise levels that could be reasonably expected fromgood experimental data.

Figure 11 and Fig. 12 show that the beta-cdf based weights are capable of maintaining all thefeatures and without any artifacts up to 220◦. The reconstruction at 200◦ is also almost distortionless.The responses are stable and the images are not affected noticeably due to the noise injection as seenin these figures.

The MAE calculated for the different weights at different angular coverages are plotted in Fig. 13and Fig. 14. The plots show that as the coverage goes further below 270◦, the beta-cdf and gamma-cdf


Figure 15. Percentage improvement of MAEfrom different weights over regular FBPP atdifferent coverages for real part of reconstructedimage.

Figure 16. Percentage improvement of MAElikewise Fig. 15 for imaginary part of recon-structed image.

weights are able to generate MAEs which remain lower than regular FBPP and are very steady up to200◦. For the beta-cdf based reconstruction, as the coverage decreases from 270◦ to 200◦, the valueof MAE increases by only 2.33% in the real part and 1.45% in the imaginary part of the image. Forgamma-cdf based reconstruction, the numbers are 4.31% and 2.64% respectively. For coverages below200◦, as the redundancy is lost, these two MAEs increase and slowly converge towards the MAE of aregular FBPP reconstruction around 180◦. For the sine-squared based weights, the MAEs are almostequal to the other two weights in the interval [250◦, 270◦], but rapidly increases as the coverage reduces,and become greater than the regular FBPP and other two MS-FBPP reconstructions. At 200◦ coverage,the MAE of the real part of the image increases by as much as 46.05% of its value at 270◦ coverage andthat of the imaginary part by 48.62%.

The MAE plots for the regular FBPP algorithm in Fig. 13 and Fig. 14 may be counterintuitiveat the first look. This is because the error increases as the angular coverage increases from 180◦ to270◦. However, it should be noted that starting from 180◦ coverage and above, there are regions in theFourier space where there is overlap from the two halves OA and OB of the semi-circular arc AOB inFig. 1. The regular FBPP doesn’t apply weights to the dataspace to account for these partial overlapsand hence gives higher errors in reconstruction. At 360◦ coverage, as there is complete coverage byboth the arcs OA and OB, weighting the dataspace is not required. Following this, it is found thaterror increases from 180◦ coverage to a maximum at 270◦, where there is maximum asymmetry withrespect to overlapping coverages in Fourier space by the two half semi-circles, OA and OB. Weightingbecomes most important at 270◦. Beyond that, the importance decreases again and vanishes at 360◦coverage. For this reason, an error increase is seen between 180◦ and 270◦ coverage. The solution tothis problem in that coverage range is to weight the projection dataspace appropriately before applyingthe backpropagation algorithm.

The percentage improvements of MAE for all the three filter classes with respect to the regularFBPP are plotted in Fig. 15 and Fig. 16. All the weights show maximum improvement at 270◦, andit decreases as the coverage gets more limited. The degradation of sine-squared filters is much greaterthan the cdf based filters. The plots show that at 180◦, with the disappearance of the redundancywithin the projection data, the MAEs for regular FBPP and that of beta-weighted MS-FBPP converge.In contrast, for the sine-squared weights, the MAE steadily increases and becomes greater than theMAE from regular FBPP. This is expected and demonstrates the effectiveness of the beta-cdf basedweights as an optimal choice. The gamma-cdf based reconstruction behaves similarly with a slightlyhigher MAE (1.66% higher than beta-cdf reconstruction in real part of image and 1.32% higher for theimaginary part of the image at 200◦ coverage).


5. CONCLUSION

In this paper, a new approach to exploit data redundancy within the traditional 2D DT setup hasbeen explored. Using cumulative distribution functions, especially the beta-cdf, it was shown thatdistortionless reconstructions of complex-valued object functions are possible even at angular coverageof 200◦. The advantages of this observation are numerous. The major benefit is that it reduces theangular scanning requirements for accurate reconstructions. This also implies shorter access times forcollecting relevant projection data. In medical applications, this can mean a lower amount of exposureto the interrogating energy and also, fewer artifacts due to temporal variations caused by movements ofthe patient. For still lower coverages (< 180◦), the redundancy in the tomographic dataset vanishes andalternate approaches such as total variation (TV) minimization, compressed sensing, etc. are exploredfor application to DT setups. Combining these techniques with MS-FBPP could be a scope of futureresearch work. In this paper, results have been validated through simulated data in the case of planewave excitation but can be extended to the fanbeam geometry as well. This is also of significance tomedical imaging. Studies with fanbeam geometry and the performance of these algorithms with realexperimental data will be explored in the future work of this study.

APPENDIX A. GAMMA-CDF BASED WEIGHT GENERATION

The gamma-cdf is defined as F (x) =∫ x−∞ f(t)dt, where f is the gamma probability density function

defined as:

f(t|a, b) =

⎧⎨⎩

1Γ(a)ba

ta−1e−t/b, t ≥ 0,

0, t < 0.(A1)

where a > 0 and b > 0 are parameters, and Γ(a) =∫ ∞0 ta−1e−tdt is standard gamma-function. As we

can see, gamma-cdf has support on [0, ∞), F (2α + π/2) < 1, which does not meet Eq. (9). Therefore,weights created using gamma-cdf in its original form will be discontinuous at the boundary between Aand B, and consequently at the boundary between B and C. One possible way to get a continuousweight is by inputting a transformed argument. So for fixed νm ∈ [−ν0, ν0], we shall construct weightsby using

w(νm, φ) = F

(tan

(π

2φ

2α + π/2

)), (A2)

where F is the gamma-cdf as defined above.

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